Macroscopic Behavior of Many-Body Quantum Systems

ended

This project is devoted to the analysis of large quantum systems. It is divided in two parts: Part A focuses on the transport properties of interacting lattice models, while Part B concerns the derivation of effective evolution equations for many-body quantum systems. The common theme is the concept of emergent effective theory: simplified models capturing the macroscopic behavior of complex systems. Different systems might share the same effective theory, a phenomenon called universality. A central goal of mathematical physics is to validate these approximations, and to understand the emergence of universality from first principles. Part A: Transport in interacting condensed matter systems. I will study charge and spin transport in 2d systems, such as graphene and topological insulators. These materials attracted enormous interest, because of their remarkable conduction properties. Neglecting many-body interactions, some of these properties can be explained mathematically. In real samples, however, electrons do interact. In order to deal with such complex systems, physicists often rely on uncontrolled expansions, numerical methods, or formal mappings in exactly solvable models. The goal is to rigorously understand the effect of many-body interactions, and to explain the emergence of universality. Part B: Effective dynamics of interacting fermionic systems. I will work on the derivation of effective theories for interacting fermions, in suitable scaling regimes. In the last 18 years, there has been great progress on the rigorous validity of celebrated effective models, e.g. Hartree and Gross-Pitaevskii theory. A lot is known for interacting bosons, for the dynamics and for the equilibrium low energy properties. Much less is known for fermions. The goal is fill the gap by proving the validity of some well-known fermionic effective theories, such as Hartree-Fock and BCS theory in the mean-field scaling, and the quantum Boltzmann equation in the kinetic scaling.